Lucas-Lehmer test is probably the easiest way to find if a Mersenne number is prime. It involves an integer sequence that states:
Let p be a prime and let Mp =2^p-1 denote a Mersenne number.
Let r1=4 be the first number in the sequence
Then rk=r(k-1)^2-2(mod Mp) and so on and so forth until k equals p-1.
Mp is prime if and only if rp-1 =0(mod Mp)
This algorithm is probably the most effective way to find if a Mersenne number is prime. Using this we can easily prove that M7 is a prime.
M7=27-1=127
R1=4
R2=42-2(mod 127)=14
R3=142-2(mod 127)=67
R4=672-2(mod 127)=42
R5=422-2(mod 127)=111
Rp-1=R6=1112-2(mod 127)=0
Therefore we can state that M7 is a prime.
Palindromic Primes
A palindrome is a sentence that reads the same in both directions backwards and forwards. A palindromic prime is a prime that reads the same backwards and forwards from a base in the center. Mersenne palindromic primes have a base 2. Since there is an equal number of digits on both sides of a single digit base, palindromic primes always have an odd number of digits. The only exception is 11 where it has an even number of digits. Here are shown are the first 11 palindromic primes where the middle number is 2.
2
30203
133020331
1713302033171
12171330203317121
151217133020331712151
1815121713302033171215181
16181512171330203317121518161
331618151217133020331712151816133
9333161815121713302033171215181613339
11933316181512171330203317121518161333911
Factorial Primes
Factorial numbers are acquired when you find the factorial of any integer. Factorial primes are when factorial numbers are prime when you add or subtract one from it. If N is any integer then a factorial prime is
N! +1
Or
N! -1
An example of a factorial prime is
6! -1
=719
Primorial Primes
A primorial is the product of primes less than or equal to an integer. Just like factorial primes, if the promorial of a number is prime when added or subtracted 1, then it is a primorial prime. When P is prime and # is the product of all the primes less than P a primorial prime is
P#+1 or P#-1
An example of a primorial prime is
11#+1
(11*7*5*3*2)+1
=2311
Prime Intervals
Prime intervals is the absolute difference between 2 nearest primes. As there is no particular pattern in the arrangement of primes, there is no particular interval between them. But observations easily show that the gap between primes increases as the primes increase in value.
Number Difference from the last number
3 ----
7 4
31 24
127 96
8191 8064
131071 122880